A359894 Number of integer partitions of n whose parts do not have the same mean as median.

0, 0, 0, 0, 1, 3, 3, 10, 13, 20, 28, 49, 53, 93, 113, 145, 203, 287, 329, 479, 556, 724, 955, 1242, 1432, 1889, 2370, 2863, 3502, 4549, 5237, 6825, 8108, 9839, 12188, 14374, 16958, 21617, 25852, 30582, 36100, 44561, 51462, 63238, 73386, 85990, 105272, 124729

Offset: 0

Author

Gus Wiseman, Jan 20 2023

Keywords

• nonn

Examples

			The a(4) = 1 through a(8) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)
         (311)   (3111)   (331)     (422)
         (2111)  (21111)  (421)     (431)
                          (511)     (521)
                          (2221)    (611)
                          (3211)    (4211)
                          (4111)    (5111)
                          (22111)   (22211)
                          (31111)   (32111)
                          (211111)  (41111)
                                    (221111)
                                    (311111)
                                    (2111111)
		

Crossrefs

The complement is counted by A240219.

These partitions are ranked by A359890, complement A359889.

The odd-length case is ranked by A359892, complement A359891.

The odd-length case is A359896, complement A359895.

The strict case is A359898, complement A359897.

The odd-length strict case is A359900, complement A359899.

A000041 counts partitions, strict A000009.

A008284 and A058398 count partitions by mean, ranked by A326567/A326568.

A008289 counts strict partitions by mean.

A027193 counts odd-length partitions, strict A067659, ranked by A026424.

A067538 counts ptns with integer mean, strict A102627, ranked by A316413.

A237984 counts ptns containing their mean, strict A240850, ranked by A327473.

A325347 counts ptns with integer median, strict A359907, ranked by A359908.

A326622 counts factorizations with integer mean, strict A328966.

A359893 and A359901 count partitions by median, odd-length A359902.

A359909 counts factorizations with the same mean as median, odd-len A359910.

Cf. A066571, A316313, A327472, A327482, A349156, A359906.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Mean[#]!=Median[#]&]],{n,0,30}]